Limit of unfair coin distribution, the hard way
Posted by peeterjoot on February 7, 2013
We calculated the distribution for the sum of random variables associated with unfair coin tosses, where the probabilities were , and for heads and tails respectively. Assigning heads and tails values of and respectively, the probability distribution of the sum of the total numbers of heads and tails values for such tosses was found to be
Part of the problem was to calculate the limit for and . I did this with the central limit theorem, but we were apparently not allowed to do it that way. Here’s a bash at it the hard way, using Stirling’s approximation and Taylor series expansion for the logs.
Application of Stirling’s approximation gives us
The term looks like it can probably be coersed into form that will allow for Taylor expansion of the log. With that change of variables, we find
This is a bit unsymmetrical, so let’s write so that
We’ve also got terms in and above that we need to express. With , we have , and
Taking logs of 1.0.2 we have
Dropping the term and noting that
This recovers the result obtained with the central limit theorem (after that result was adjusted to account for parity).